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### Two Mathematicians Problem

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Date: 05/18/98 at 04:46:56
From: Yusuf Kursat TUNCEL
Subject: 2 Mathematicians Problem

Dear Dr. Math,

We are given that X and Y are two integers, greater than 1, are not
equal, and their sum is less than 100. A and B are two talented
mathematicians; A is given the sum, and B is given the product of
these numbers:

B says "I cannot find these numbers."
A says "I was sure that you could not find them."
B says "Then, I found these numbers."
A says "If you could find them, then I also found them."

What are these numbers?

Here is my approach:

The reason that B cannot find the numbers from their products is that
the product has more than 2 prime factors that are not the same (like
2 * 3 * 5, which results in products like 10 * 3, 2 * 15, and 5 * 6)
So, for A to be sure that B could not find the numbers, A must have a
sum that does not contain a two prime solution pair such as
2 + 13 = 15. A cannot have this because he wouldn't be sure if the
product is given 2 * 13 = 26 or not.

I have written all of the prime numbers from 2 to 100, and added them
with each other, I have eliminated the the results from the set of
numbers 2 - 100 (also 3, 4, 5, 6, 7, 8, 9, and 10 can not be sum and
are eliminated automatically) so that I have a set, which contains
probable sum of the numbers:

{11, 17, 23, 27, 29, 33, 35, 37, 41, 47, 51, 53, 57,  59,  65, 67,
71, 75, 77, 79, 83, 87, 89, 93, 95, 97, 98}

Any of the numbers in this set cannot be constructed just by using two
prime numbers. But I couldn't go far. Would you help me?
```

```
Date: 05/18/98 at 17:29:46
From: Doctor Rob
Subject: Re: 2 Mathematicians Problem

Your approach is the right one, but you must be careful.

First of all:

2 + 2 = 4 <= X + Y <= 99   and
2 * 2 = 4 <= X * Y <= 2450 = 49 * 50

B says, "I can not find these numbers."

Then X * Y cannot be prime, since it is the product of two numbers
greater than 1. It also cannot be the square of a prime number,
because X and Y are not equal. If X * Y had exactly two proper
divisors, then B would know the two numbers. This eliminates the
product of two distinct primes, and the cube of any prime.

A says, "I was sure that you could not find them."

Yes, X + Y cannot be the sum of two distinct primes, or the sum
of a prime and its square. This forces X + Y to be one of the
following numbers:

11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67,
71, 77, 79, 83, 87, 89, 93, 95, 97

Since all the remaining sums are odd, one of the two numbers must be
odd and the other even. X * Y must then be even.

B says, "Then, I found these numbers."

B can throw away any factorization of X * Y in which the sum of the
two factors is not one of these numbers. There must be exactly one
such possibility left for him to know the numbers. This eliminates
X * Y = 12, for example, because even though it has factorizations
2 * 6 and 3 * 4, the sums of the two factors, 8 and 7 respectively,
are not in the set of possible sums, so there are no possibilities
left. It also throws out X * Y = 120, because 120 = 5 * 24 = 15 * 8,
and 5 + 24 = 29 and 15 + 8 = 23, which are both on the list, so there
are two possibilities left. This limits the possibilities for the
product to:

18 = 9 * 2, 9 + 2 =     11
24 = 8 * 3, 8 + 3 =     11
28 = 7 * 4, 7 + 4 =     11
50 = 25 * 2, 25 + 2 =   27
52 = 13 * 4, 13 + 4 =   17
54 = 27 * 2, 27 + 2 =   29
76 = 19 * 4, 19 + 4 =   23
92 = 23 * 4, 23 + 4 =   27
96 = 32 * 3, 32 + 3 =   35
98 = 49 * 2, 49 + 2 =   51
100 = 25 * 4, 25 + 4 =   29
112 = 16 * 7, 16 + 7 =   23
124 = 31 * 4, 31 + 4 =   35
140 = 35 * 2, 35 + 2 =   37
144 = 48 * 3, 48 + 3 =   51
148 = 37 * 4, 37 + 4 =   41
152 = 19 * 8, 19 + 8 =   27
160 = 32 * 5, 32 + 5 =   37
172 = 43 * 4, 43 + 4 =   47
176 = 16 * 11, 16 + 11 = 27
188 = 47 * 4, 47 + 4 =   51
192                      67
198                      29
208                      29
212                      57
216                      35
220                      59
222                      77
228                      79
230                      51
232                      37
234                      35
238                      41
244                      65
246                      47
250                      35
...                      ...

A says, "If you could find them, then I also found them."

This means that the sum in the last preceding list must occur only
once. That eliminates X + Y = 11, 27, 29, 23, 35, 51, 37, 41, 47, ...,
leaving only a single sum that occurs only once in the above table.
This tells you X and Y and how A and B figured them out.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
High School Logic
High School Puzzles
Middle School Logic
Middle School Puzzles

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